Rick Blute - Nuclear ideals

Closed categories, either monoidal or cartesian, provide the foundation for the categorical modelling of logics, such as linear or intuitionistic logic. In this work, I will introduce a new categorical construction, the nuclear ideal. These exist within monoidal categories in which only certain of the morphisms allow the sort of transpositions implied by the existence of a closed structure. Examples arise in the category of Hilbert spaces, and a category of distributions on Euclidean space.

This work can also be viewed as a generalization of the category of relations. Indeed the original motivation for this study came from attempts to generalize the structure of the category of relations to handle what might be called ``probabilistic relations'' with an eye towards certain applications in computer science.

We also extend the recent work of Joyal, Street and Verity on traced monoidal categories to this setting by introducing the notion of a trace ideal. Thus our work can be viewed as modelling a generalized form of Girard's Geometry of Interaction, an extremely novel approach to the semantics of proof theory designed to capture the dynamics of normalization.

Finally, we will mention some of the possible applications of nuclear ideals to the categorical structures arising from topological quantum field theory and conformal field theory.

This talk will summarize joint work with S. Abramsky, P. Panangaden and D. Pronk.